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A128741 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k returns to the x-axis (1 <= k <= n).

%I A128741 #8 Jul 23 2017 12:22:33
%S A128741 1,2,1,6,3,1,20,11,4,1,72,42,17,5,1,274,166,72,24,6,1,1086,675,307,
%T A128741 111,32,7,1,4438,2809,1322,506,160,41,8,1,18570,11913,5752,2296,775,
%U A128741 220,51,9,1,79174,51319,25274,10418,3692,1127,292,62,10,1,342738,223977,112054
%N A128741 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k returns to the x-axis (1 <= k <= n).
%C A128741 A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.
%C A128741 Row sums yield A002212.
%C A128741 T(n,1) = 2*A002212(n-1) for n >= 2 (obvious: the path of semilength n with exactly one return are of the form UPD and UPL, where P is a path of semilength n-1).
%C A128741 Sum_{k=1..n} k*T(n,k) = A128742(n).
%H A128741 E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
%F A128741 G.f.: (1 - tz + tzg)/(1 - tzg) - 1, where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).
%F A128741 Column k has g.f. z^k*g^(k-1)*(2g-1).
%e A128741 T(4,3)=4 because we have U(D)U(D)UUD(D), U(D)U(D)UUD(L), U(D)UUD(D)U(D) and UUD(D)U(D)U(D) (the return steps to the x-axis are shown between parentheses).
%e A128741 Triangle starts:
%e A128741    1;
%e A128741    2,  1;
%e A128741    6,  3,  1;
%e A128741   20, 11,  4,  1;
%e A128741   72, 42, 17,  5,  1;
%p A128741 g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-t*z+t*z*g)/(1-t*z*g)-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n),n=1..11) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od;
%Y A128741 Cf. A002212, A128742.
%K A128741 tabl,nonn
%O A128741 1,2
%A A128741 _Emeric Deutsch_, Mar 31 2007